cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).

Original entry on oeis.org

1, 1, 2, 1, 2, 8, 2, 1, 2, 4, 2, 16, 2, 8, 16, 1, 2, 24, 2, 24, 16, 8, 2, 64, 2, 4, 8, 16, 2, 1024, 2, 3, 16, 4, 16, 48, 2, 8, 16, 48, 2, 2048, 2, 48, 96, 8, 2, 128, 6, 12, 16, 8, 2, 768, 16, 128, 16, 4, 2, 147456, 2, 8, 32, 3, 16, 2048, 2, 24, 16, 1024, 2
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2020

Keywords

Examples

			a(6) = gcd(tau(1), sigma(1)) * gcd(tau(2), sigma(2)) * gcd(tau(3), sigma(3)) * gcd(tau(6), sigma(6)) = gcd(1, 1) * gcd(2, 3) * gcd(2, 4) * gcd(4, 12) = 1 * 1 * 2 * 4 = 8.

Links

Crossrefs

Cf. A334491 (Product_{d|n} gcd(d, sigma(d))), A334579 (Sum_{d|n} gcd(tau(d), sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).

Programs

  • Magma
    [&*[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Maple
    g:= proc(d) option remember; igcd(numtheory:-tau(d), numtheory:-sigma(d)) end proc:
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, May 11 2020
  • Mathematica
    a[n_] := Product[GCD[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), sigma(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

a(p) = 2 for p = odd primes (A065091).

A334490 a(n) = Sum_{d|n} gcd(d, sigma(d)).

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 5, 2, 14, 2, 5, 6, 5, 2, 13, 2, 8, 4, 5, 2, 27, 3, 5, 4, 34, 2, 21, 2, 6, 6, 5, 4, 19, 2, 5, 4, 19, 2, 19, 2, 10, 10, 5, 2, 32, 3, 7, 6, 8, 2, 20, 4, 43, 4, 5, 2, 40, 2, 5, 6, 7, 4, 21, 2, 8, 6, 11, 2, 35, 2, 5, 8, 10, 4, 19, 2, 22
Offset: 1

Views

Author

Jaroslav Krizek, May 03 2020

Keywords

Examples

			a(6) = gcd(1, sigma(1)) + gcd(2, sigma(2)) + gcd(3, sigma(3)) + gcd(6, sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(3, 4) + gcd(6, 12) = 1 + 1 + 1 + 6 = 9.

Links

Crossrefs

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A000203 (Sum_{d|n} gcd(d, pod(d)) = sigma(n)).
Cf. A000040, A007955 (pod(n)), A334491 (for product instead of sum).
Inverse Möbius transform of A009194.

Programs

  • Magma
    [&+[GCD(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := DivisorSum[n, GCD[#, DivisorSigma[1, #]] &]; Array[a, 80] (* Amiram Eldar, May 03 2020 *)
  • PARI
    a(n) = sumdiv(n, d, gcd(d, sigma(d))); \\ Michel Marcus, May 03 2020

Formula

a(p) = 2 for p = primes (A000040).
a(n) = Sum_{d|n} A009194(d). - Antti Karttunen, May 09 2020

A334664 a(n) = Product_{d|n} gcd(d, tau(d)).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 8, 3, 4, 1, 24, 1, 4, 1, 8, 1, 72, 1, 8, 1, 4, 1, 768, 1, 4, 3, 8, 1, 16, 1, 16, 1, 4, 1, 3888, 1, 4, 1, 256, 1, 16, 1, 8, 9, 4, 1, 1536, 1, 8, 1, 8, 1, 144, 1, 256, 1, 4, 1, 2304, 1, 4, 9, 16, 1, 16, 1, 8, 1, 16, 1, 1492992, 1, 4, 3, 8, 1
Offset: 1

Views

Author

Jaroslav Krizek, May 07 2020

Keywords

Examples

			a(6) = gcd(1, tau(1)) * gcd(2, tau(2)) * gcd(3, tau(3)) * gcd(6, tau(6)) = gcd(1, 1) * gcd(2, 2) * gcd(3, 2) * gcd(6, 4) = 1 * 2 * 1 * 2 = 4.

Links

Crossrefs

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009191 (gcd(n, tau(n))).

Programs

  • Magma
    [&*[GCD(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Table[Times@@GCD[Divisors[n],DivisorSigma[0,Divisors[n]]],{n,80}] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(d[k], numdiv(d[k]))); \\ Michel Marcus, May 08-11 2020

Formula

a(p) = 1 for p = odd primes (A065091).

A334805 a(n) = Product_{d|n} lcm(d, sigma(d)) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 168, 30, 864, 56, 20160, 1404, 16200, 132, 2032128, 182, 56448, 43200, 9999360, 306, 23654592, 380, 190512000, 451584, 313632, 552, 29262643200, 23250, 596232, 1516320, 88510464, 870, 100776960000, 992, 20158709760, 836352, 1685448, 2822400
Offset: 1

Views

Author

Jaroslav Krizek, Jun 26 2020

Keywords

Examples

			a(6) = lcm(1, sigma(1)) * lcm(2, sigma(2)) * lcm(3, sigma(3)) * lcm(6, sigma(6)) = lcm(1, 1) * lcm(2, 3) * lcm(3, 4) * lcm(6, 12) = 1 * 6 * 12 * 12 = 864.

Crossrefs

Cf. A334783 (Sum_{d|n} lcm(d, sigma(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).
Cf. A000203 (sigma(n)), A009242 (lcm(n, sigma(n))), A036690.

Programs

  • Magma
    [&*[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[LCM[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

Formula

a(p) = p^2 + p for p = primes (A000040).
Showing 1-4 of 4 results.

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